The fabrication of semiconductor integrated circuits begins with a semiconductor wafer (e.g., silicon) which is often referred to as the "starting material." Typically the starting material is a lightly doped p-type wafer (with an approximate dose of about 5.times.10.sup.14 -1.times.10.sup.15 atoms/cm.sup.2) having a &lt;100&gt; crystal orientation. It is desirable that the doping concentration (atoms/cm.sup.3) be relatively uniform across a single wafer (e.g., within about 1%) and be within a range of about 10.sup.14 to 10.sup.16 atoms/cm.sup.3 and be uniform from wafer to wafer to provide a resistivity of about 2 to 100 .OMEGA.-cm. One of the reasons that the doping level uniformity is important is that the doping level of the wafer (i.e., the substrate for various semiconductor devices) impacts the resultant threshold voltage for transistors built on and in the substrate as well as other transistor device parameters.
When the source and substrate of a transistor are shorted together (V.sub.SB =0), the transistor threshold voltage (V.sub.T) can be characterized by the following equation: EQU V.sub.T =V.sub.TO =.phi..sub.ms -2.phi..sub.f -Q.sub.tot /C.sub.OX -Q.sub.BO /C.sub.OX,
wherein .phi..sub.ms is the work function difference between the gate material and the bulk silicon in the transistor channel, .phi..sub.f is the equilibrium electrostatic potential, Q.sub.BO is the charge stored per unit area (C/cm.sup.2) in the depletion region, C.sub.OX is the gate oxide capacitance per unit area (F/cm.sup.2), and Q.sub.tot is the total positive oxide charge per unit area present at the oxide/bulk interface. .phi..sub.ms, .phi..sub.f and Q.sub.BO are each dependent upon the doping concentration of the starting material, wherein an increase in the doping concentration of the substrate changes the above parameters and results in an increase in the threshold voltage (V.sub.T). Consequently, nonuniform doping concentrations within a wafer and between wafers is undesirable since it results in variations in the resulting transistor threshold voltages for various semiconductor devices.
In addition to it being desirable to provide starting material which provides uniform transistor threshold voltages within a wafer and between wafers, it is also desirable for the starting material to provide a low source/drain-to-substrate capacitance, a high source/drain-to-substrate breakdown voltage, high current mobility and a low sensitivity to source-substrate bias effects. A deviation in the desired doping concentration, however, impacts the above characteristics. For example, an increase in the wafer doping concentration undesirably results in a decrease in the junction breakdown voltage, increases the junction capacitance and lowers the carrier mobility.
Because the wafer doping concentration level and uniformity is an important characteristic, electrical specifications are provided for the starting material which include, for example: the conductivity type of the wafer, the average resistivity or resistivity range (.OMEGA.-cm), the radial resistivity gradient (% variation) and resistivity variations. The conductivity type information includes whether the wafer is an n-type or a p-type wafer and indicates what element was used to dope the wafer (e.g., arsenic, phosphorous, boron, etc.). The wafer resistivity relates to the doping density of the wafer (atoms/cm.sup.3) and is measured using a four-point probe technique. The radial resistivity gradient provides a measure of the variation of the resistivity between the center and selected outer regions of the wafer, and the resistivity variations represent local variations of the resistivity on the wafer. Both the radial resistivity and the resistivity variations are also measured using a four-point probe technique.
The four-point probe technique is used to measure the sheet resistance Rs of a film. The sheet resistance Rs of a film (which in this instance is the wafer) is determined as follows in conjunction with prior art FIG. 1. The resistance (R) of a rectangular shaped film of length (L), width (W) and thickness (t) is given by the equation: EQU R=.rho.L/tW,
wherein .rho. equals the resistivity of the film, which is unique for a given material, and is measured in .OMEGA.-cm. If the length L is equal to the width W, then the rectangle is a square and the equation reduces to:
R=.rho./t=Rs,
wherein Rs is the sheet resistance in .OMEGA./square and is independent of the size of the square (but does depend on the resistivity of the material and the thickness of the film). Therefore the resistivity .rho. and the sheet resistance Rs are distinct parameters that are related by the above equation.
The four-point probe method is illustrated in prior art FIG. 2. If the sample film may be approximated as semi-infinite with respect to the spacings (s) between the four probes (which are spaced apart substantially equally from one another), the current (I) is driven as shown and the voltage drop (V.sub.1 -V.sub.2) is measured across the remaining probes as illustrated in prior art FIG. 2. The sheet resistance may then be calculated according to the following equation: EQU Rs=(V.sub.1 -V.sub.2)(2.pi.s)/It.
To prevent erroneous readings using the four point technique (e.g., due to thermoelectric heating and cooling) the measurement is often performed with current forced in both directions and the two readings are averaged. Further, the test is often performed at several current levels (i.e., I.sub.1, I.sub.2, etc.), until the proper current level is found. For example, if the current is too low, the forward and reverse current readings will substantially differ and if the current is too high, I.sup.2 R heating will result in the measured reading drifting over time. Although the American Society for Testing and Materials (ASTM) provides standards which recommend current levels for a given resistivity range, one may still need to vary the current about the recommended value to achieve the optimum current for an accurate measurement which undesirably takes extra time.
The sheet resistance Rs and the resistivity .rho. are found using the measured results and the equation V/I(2.pi.s), wherein s is the probe spacing. The above equation, however, is only accurate if the sample is semi-infinite with respect to the probe spacings, which is often not an accurate assumption. Thus, the sheet resistance is typically calculated by the relation: EQU Rs=(V/I)F.sub.1,
wherein F.sub.1 is a correction factor which is a function of the average probe distance s and the wafer diameter D (i.e., F.sub.1 =f(s/D)).
Since the four-point probe technique uses a correction factor and occupies a space of at least 3s due to the four probes, the readings are not totally accurate and further represent merely an average resistivity within the region of 3s. Consequently, it would bee desirable to have a method of determining the resistivity of the starting material that provides a more accurate and convenient resistivity reading and a higher resolution mapping of the resistivity across the wafer to thereby determine whether the doping level concentration is sufficiently uniform.